3.

4.

5.

1.

2.

Day 3: Daily Warm-up.

22x

33 xx

1235 xx

Find the product and combine like terms.

323149 2424 xxxx

xxxxxx 3135 2323

Simplify each expression (combine like terms)

A.3a Polynomials and FactoringSpecial Products and Factoring

A.3 ObjectivesA. Polynomials and Factoring

1. Understand the vocabulary of polynomials

2. Add and subtract polynomials

3. Write polynomials in standard form.

4. Multiply polynomials

5. Special Products

6. GCF

7. Factor Polynomials with 2 or 3 terms.

8. Factor Polynomials with more than 3 terms.

9. Find the GCF of any polynomial expression

1. Vocabulary of Polynomials

the product of a constant and a variable raised to a nonnegative integer power.

ax kcoefficient of the monomial (constant value) variable

degree of the monomial (power of the variable)

Monomial:

Polynomial: sum or difference of monomials

3x 2 4y 2x 6 5x 2 3y 2 3x 1

1. Vocabulary of Polynomials

Special Polynomial Namesif # terms is: Name: 2 terms Binomial 3 terms Trinomial

Degree of Polynomialif Highest degree term is: Name:

degree 1Linear

degree 2Quadratic

degree 3Cubic

Like terms : contain same power of the variable.

2. Add/Subtract Polynomials

323149 1) 2424 xxxx

Add the coefficients of like terms(same power of the variable)

xxxxxx 3135 2) 2323

Examples

3. Standard Form and Degree

Standard Form:

3x x 4 4x 2 7

polynomial written in descending order

Definition

Example

Write the standard form for:

The Degree of a polynomial: is the degree of the greatest degree term.Definition

Example

What is the degree of this polynomial? 24x

4. Multiplying Polynomials

1423 2 xxx

• Special case: only works with two binomials2. Foil Method

1. Double Distribution• Using the distributive property, we can multiply any

number of terms

23 xx

4. Multiplying PolynomialsExamples for YOU to try….

11 1) 2 xxx

yxyx )13()13( 2)

xxx 1212 3)

Find the product and write in STANDARD form

5. Special Products

BABA

These are products that occur often. You should know these!Study Tip

Difference of Two Squares.

2BA

Perfect Square Binomial.

2BA

22 BA

22 2 BABA

22 2 BABA

6. GCF (Greatest Common Factor)

xxx 1624 23

GCF of coefficients: Largest number that divides into all coefficients.GCF of variable expressions: Find smallest exponent

Example. Factor out the GCF

Definition of Prime: If a polynomial does not factor into 2 or more polynomials, it is Prime.

3.

4. Factor completely.

1.

2. State the domain of

Day 4: Daily Warm-up.

22

2323

6

1

zyx

zyx

xxx 36123 23

Simplify (no negative exponents)

22 11 xx

42

1

x

Simplify completely.

7. Factoring PolynomialsDefinition: Factoring is writing a polynomial as a

product of polynomials of lower degree

General Steps for Factoring:

1. Factor out GCFs2. Is it a binomial (with no middle term) ?

a) Is it a Difference of squares or sum/difference of cubes or or

3. Is it a trinomial (only 3 terms)• apply factorization algorithm

4. If more than 3 terms• grouping

22 BA 33 BA 33 BA

5. More Special Products

22 BABABA

Difference of Two Cubes.

Sum of Two Cubes.

22 BABABA

33 BA

33 BA

A. Factoring Binomials

Factor each. If it does not factor, state that it is PRIME.1.

2.

3.

4.

5.

Study Tip

252 x

254 4 x

1253x

8x 3 1

12 x

B. Factoring Trinomials: Simple case, when

1. Factor out GCF, if there is one.

cbxax 2

1242 xx

2. What multiplies to c and adds up to b?

c b

3. Rewrite as:

Example 0.

1a

) )( ( xx

C. Factoring Algorithm for case when

1. Factor out GCF.

cbxax 2

310 2 xx

2. Multiply a and c

3. Find the factors of ac that add to b.

4. Rewrite the middle term bx as sum of the 2 factors.

5. Grouping.-Double bubble (check signs)-Factor out GCF in each group, if no GCF, write 1.

Example 1.

30)3(10 ac 1b

1a

Factoring Algorithm

1. Factor out GCF.

cbxax 2

7196 2 xx

2. Multiply a and c3. Find the factors of ac that add to b.

4. Rewrite the middle term bx as sum of the 2 factors.5. Grouping.

-Double bubble (check signs)

-Factor out GCF in each group

Example 2.

ac b

Factoring Algorithm for simple case,

1. Factor out GCF.

cbxx 2

1242 xx

2. What multiplies to c and adds up to b?

c b

3. We can skip rewrite of the middle term. (Why?)

Example 3.

1a

) )( ( xx

Factor this polynomial using the “double bubble” algorithm. Do you get the same result?

Practice Time!completely factor the polynomial

12 36 xx

30328 2 xx

1.

3.

4.

2.

57 xx

25 xx

7. Polynomial with 4 terms

3x 3 2x 2 12x 8Ex.

Use Grouping (double bubble)

Factoring Practice completely factor the polynomial

20x 6 12x 5 35x 3 21x 2

4x 3 20x 2 9x 451.

2.

324 29)2(6 xxxx 1.

8. Finding the GCF of an expression8. Finding the GCF of an expression

2. 433243 2 xxx

5. More Special Products

3BA

Cubes of Binomials, or Perfect Cubes.

3BA

3223 33 BABBAA

3223 33 BABBAA

You may wish to memorize these, but could also derive them.